When it comes to learning about geometry and shapes, one of the most important concepts to understand is that of a parallelogram. A parallelogram is a quadrilateral with two pairs of parallel sides, which means that if you were to draw a line parallel to one side of the shape, it would intersect both pairs of opposite sides at the same point.

In this article, we will explore the properties and characteristics of parallelograms, including how to find their area. But first, let's take a look at an example of a parallelogram ABCD, where E and F are the mid-points of sides AB and CD respectively.

As you can see from the diagram, the line segments AF and EC trisect the diagonal BD. This means that each of these lines divides the diagonal into three equal parts. But how do we know this to be true?

To prove that the line segments AF and EC trisect the diagonal BD, we need to show that they are equal in length. To do this, we can use the fact that a parallelogram is a quadrilateral with two pairs of parallel sides.

Let's start by drawing a line parallel to side AB through point X. This line will intersect both sides AC and CD at points P and Q respectively. As we know from the definition of a parallelogram, these lines are parallel to each other.

Now let's focus on quadrilateral PQC. Since PQ is parallel to AB, we can conclude that QC is also parallel to AB. Similarly, since QF is parallel to CD, we can conclude that PF is also parallel to CD.

But this means that quadrilaterals PPF and QQC are similar, since they both have two pairs of parallel sides. This in turn means that the corresponding sides of these two quadrilaterals are proportional.

In particular, the ratio of PQ to QC is equal to the ratio of AB to BC. Since we know that AB = 24 cm, we can conclude that QC is also equal to 24 cm.

But this means that the line segment AF is equal in length to the line segment EC, since both are parallel to each other and intersect at right angles. Therefore, we can conclude that the line segments AF and EC trisect the diagonal BD.

Now that we have proven this property of parallelograms, let's move on to find the area of the shape. To do this, we need to use the formula for the area of a parallelogram, which is base times height.

In our example, the base of the triangle is segment ML, which has a length of 24 cm. The height of the triangle is also given as 7 cm. Therefore, we can plug in these values into the formula and calculate the area of the triangle.

Area = base x height

= 24 cm x 7 cm

= 168 square centimeters

Therefore, the area of the parallelogram ABCD is 168 square centimeters. But what does this mean?

In practical terms, the area of a shape can be used to calculate its size or volume. For example, if you were designing a room with a rectangular floor plan, you would need to know the area of the room in order to determine how many tiles you would need to cover it.

But that's not all - the area of a shape can also be used to calculate its perimeter or circumference. For example, if you were designing a circular garden bed, you would need to know the area of the circle in order to determine how much soil and plants you would need.

In conclusion, parallelograms are an important type of quadrilateral with many practical applications. By understanding their properties and characteristics, including how to find their area, we can apply this knowledge to solve real-world problems and challenges.

Parallelogram abcd